Related papers: Imaginaries and definable types in algebraically c…

We study interpretable sets in henselian and sigma-henselian valued fields with value group elementarily equivalent to Q or Z. Our first result is an Ax-Kochen-Ershov type principle for weak elimination of imaginaries in finitely ramified…

Schunck classes, saturated formations and projectors were originally defined for finite soluble groups for which they provided families of intravariant subgroups. In this paper, I set out the analogous theory for finite-dimensional soluble…

Classification and invariants, with respect to basis changes, of finite dimensional algebras are considered. An invariant open, dense (in the Zariscki topology) subset of the space of structural constants is defined. The algebras with…

We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and…

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…

We prove a tight bound on the number of realized $0/1$ patterns (or equivalently on the Vapnik-Chervonenkis codensity) of definable families in models of the theory of algebraically closed valued fields with a non-archimedean valuation. Our…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (a)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

We introduce and study a new class of differential fields in positive characteristic. We call them separably differentially closed fields and demonstrate that they are the differential analogue of separably closed fields. We prove several…

We give a construction of a large first-order definable family of subrings of finitely generated fields $K$ of any characteristic. We deduce that for any such $K$ there exists a first-order sentence $\varphi_K$ characterising $K$ in the…

We develop the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated…

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (=a finite union of hyperplanes) whose Levi-Civita connection is of Dunkl…

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…

A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.

We consider the problem of characterizing isomorphisms of types, or, equivalently, constructive cardinality of sets, in the simultaneous presence of disjoint unions, Cartesian products, and exponentials. Mostly relying on results about…

We show that all extended functorial field theories, both topological and nontopological, are local. We define the smooth (infinity,d)-category of bordisms with geometric data, such as Riemannian metrics or geometric string structures, and…

We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups,…

Shape grammars compute over shapes which are defined in the universe $U^*$. Shapes in the universe $U^*$ are analogous to line drawings that can be physically realized in the plane. Any shape is embedded or contained in an arrangement of…

We investigate a class of nominal algebraic Henkin-style models for the simply typed lambda-calculus in which variables map to names in the denotation and lambda-abstraction maps to a (non-functional) name-abstraction operation. The…

Motivated by the work of J\"urgen Neukirch and Ivan Fesenko we propose a general definition of an abelian class field theory from a purely group-theoretical and functorial point of view. This definition allows a modeling of abelian…

We develop an explicit theory of formal modular forms over arbitrary number fields $K$, as functions of modular points. We define modular points for $\Gamma_0({\mathfrak n})$ and $\Gamma_1({\mathfrak n})$, where the level ${\mathfrak n}$ is…